Area of the circular segment bounded by a hexagon's side

Text

In a circle of radius $r=10$ cm consider the chord equal to the side of the inscribed regular hexagon. Compute the area of the circular segment bounded by that chord and its (minor) arc.

Solution

The side of the inscribed regular hexagon subtends a central angle of $60^\circ$. Area of the $60^\circ$ sector: $\ \dfrac{60}{360}\pi r^2=\dfrac{\pi r^2}{6}=\dfrac{100\pi}{6}=\dfrac{50\pi}{3}$. Area of the triangle with sides $r,r$ and angle $60^\circ$: $\ \dfrac12 r^2\sin60^\circ=\dfrac12\cdot100\cdot\dfrac{\sqrt3}{2}=25\sqrt3$. Segment $=\dfrac{50\pi}{3}-25\sqrt3\approx52.36-43.30=9.06\ \text{cm}^2.$ $\boxed{\mathcal A=\tfrac{50\pi}{3}-25\sqrt3\approx9.06\ \text{cm}^2}$